To determine how much tax Noam and Betty owe based on their taxable incomes and the given tax brackets, let's carefully go through the calculation.
### For Noam:
Noam's taxable income is \[tex]$90,000. This falls in the fourth bracket, where \(85,526 \leq 90,000 \leq 163,300\), so we use the third condition:
\[
T(d) = 0.24(d - 85,525) + 14605.5
\]
Plugging in \(d = 90,000\):
\[
T(90,000) = 0.24(90,000 - 85,525) + 14,605.5
\]
\[
T(90,000) = 0.24(4,475) + 14,605.5
\]
\[
T(90,000) = 1,074 + 14,605.5
\]
\[
T(90,000) = 15,679.5
\]
So, Noam owes \$[/tex]15,679.5 in taxes.
### For Betty:
Betty's taxable income is \[tex]$120,000. This falls in the fourth bracket as well, where \(85,526 \leq 120,000 \leq 163,300\), so we use the same fourth condition:
\[
T(d) = 0.24(d - 85,525) + 14605.5
\]
Plugging in \(d = 120,000\):
\[
T(120,000) = 0.24(120,000 - 85,525) + 14,605.5
\]
\[
T(120,000) = 0.24(34,475) + 14,605.5
\]
\[
T(120,000) = 8,274 + 14,605.5
\]
\[
T(120,000) = 22,879.5
\]
So, Betty owes \$[/tex]22,879.5 in taxes.
### Difference in Tax:
To find out how much more tax Betty pays compared to Noam, we find the difference between their tax amounts:
[tex]\[
\text{Tax Difference} = T(120,000) - T(90,000)
\][/tex]
[tex]\[
\text{Tax Difference} = 22,879.5 - 15,679.5
\][/tex]
[tex]\[
\text{Tax Difference} = 7,200
\][/tex]
Thus, Betty pays \$7,200 more in taxes than Noam.
